Efficient and Versatile Model for Vibrational STEM-EELS

Abstract

We introduce a novel method for the simulation of the impact scattering in vibrational scanning transmission electron microscopy electron energy loss spectroscopy (STEM-EELS) simulations. The phonon-loss process is modeled by a combination of molecular dynamics and elastic multislice calculations within a modified frozen phonon approximation. The key idea is thereby to use a so-called δ-thermostat in the classical molecular dynamics simulation to generate frequency dependent configurations of the vibrating specimen's atomic structure. The method includes correlated motion of atoms and provides vibrational spectrum images at the cost comparable to standard frozen phonon calculations. We demonstrate good agreement of our method with simulations and experiments for a 15 nm flake of hexagonal boron-nitride (hBN).Excellent spatial resolution is the highlight of the transmission electron microscope [1-6], allowing atomically resolved elemental mapping and chemistry. Recent instrumental developments have improved the energy resolution of electron energy loss spectroscopy down to 4.2 meV [7,8]. This unique combination of high spatial and energy resolution opened doors to unprecedented experiments such as temperature measurement at the nanoscale [9, 10], identification and mapping of isotopically labeled molecules [11], position-and momentum-resolved mapping of phonon modes [12,13], mapping of bulk and surface modes of nanocubes [14], or investigations of the nature of polariton modes in van der Waals crystals [15].Inelastic electron scattering on atomic vibrations in a polar material consists of two major contributions, namely, impact scattering and dipolar scattering [16,17]. The latter stems from a long ranged interaction between the beam electron and an oscillating dipole moment generated by the atomic vibrations. It can be used to perform damage-free vibrational spectroscopy in an aloof beam geometry [18,19]. High-angle impact scattering on the other hand is quite localized scattering on the atomic potential and allows for high resolution imaging and spectroscopy [17,[20][21][22].From a theoretical point of view, there is a limited number of approaches one can follow to simulate vibrational spectroscopy and imaging. Dwyer presents a treatment of both single impact and single dipolar scattering including a detailed phonon dispersion from a density functional theory calculation in a multislice scheme [23]. Forbes et al. introduced a quantum excitation of phonons model, which allows for the inclusion of thermal diffuse scattering into multislice image simulations [24]. The thermal vibrations are modeled within the Einstein model for quick image calculations, neglecting correlations in atomic movements and dispersion effects. Such an approach is not able to simulate vibrational spectra but in principle one could use different models of atomic vibrations to generate the necessary configurations. In a more recent work, Forbes et al. described a method to model the vibrational sector of the electron energy loss spectrum [25...

Contrast reversal in atomic-scale phonon spectroscopic imaging

“…The QEP model results in Fig. 2(d) were calculated using an Einstein model-a very good approximation which yields the same integrated spectrum as that obtained in a correlated model [19], where Debye-Waller factors were defined in terms of mean square thermal displacements of 0.007 Å 2 for B and 0.011 Å 2 for N. The integrity of this procedure was confirmed by Zeiger and Rusz [20], who showed explicitly that a more detailed approach that includes the correlated motion of atoms based on molecular dynamics yields results very similar to the Einstein model for such spectral maps. The simulated maps in Figs.…”

The frequency resolved frozen phonon method for vibrational STEM-EELS

“…We have recently introduced a versatile and efficient model for vibrational STEM-EELS [10], which we shall call "frequency-resolved frozen phonon multislice method" (FRFPMS). The essence of the method is, that a colored-noise thermostat based on the generalized Langevin equation is used to excite a nonequilibrium distribution of velocities in a molecular dynamics (MD) simulations in such a way, that modes, whose frequency lies within a narrow interval, are heated, while most other modes are kept at a much lower temperature [11,12,13].…”

“…Each of the inward facing tips of the equilateral triangles cut out of the diffraction patterns is located at the (000) position in the diffraction pattern. The structure model and MD potential used for this figure is the same as in Ref [10],. but here parallel illumination was considered instead of a convergent beam.…”

Bridging nano-optics and condensed matter formalisms in a unified description of inelastic scattering of relativistic electron beams

“…On the other hand, employing the relativistic MDFF to describe low-loss spectroscopy experiments directly paves the way toward the ab-initio modelling of EELS experiments on nano-optical systems, with potential and far-reaching applications in, e.g., quantum plasmonics [100] or excitonplasmon [101,102] and phonon-plasmon coupling physics [101,103]. Also, this should ease bridging the antagonist descriptions of EELS phonon spectroscopy experiments, either quantum [104][105][106][107] or classical, in the quasi-static [108,109] or retarded [110] regimes, or attempts to mix the two [111]. Besides, by putting the relativistic MDFF at the core of our approach, we enable the extensive use of ab-initio methods for the modelling of EELS experiment, in the same vein as the pioneering works employing density functional theory [112,113].…”